Maharashtra Board Class 10 Maths Part I Chapter 2 Quadratic Equations PDF Download

Quadratic Equations

This chapter will teach you about quadratic equations. You will learn what they are and how to solve them. You will also learn about the nature of roots and how roots relate to coefficients. Finally, you will see how to use quadratic equations to solve real-life problems.

Teacher's Note

A quadratic equation is any equation where the highest power of x is 2. For example, when you buy a rectangular plot of land, you use quadratic equations to find its length and breadth. This is very common in India when buying agricultural land.

Exam Trick

Remember: Quadratic means "square" - so look for \(x^2\) in the equation. If there is no \(x^2\), it is not a quadratic equation. Write the equation in the form \(ax^2 + bx + c = 0\) to make solving easier.

Points to Remember

A quadratic equation has the highest power of the variable as 2.

The general form is \(ax^2 + bx + c = 0\) where a, b, and c are real numbers.

The value of a must never be zero.

There are three main ways to solve a quadratic equation: factorisation, completing the square, and using the formula.

What Is A Quadratic Equation?

You learned about polynomials last year. A polynomial with degree 1 is called linear. A polynomial with degree 2 is called quadratic.

When you put a quadratic polynomial equal to zero, you get a quadratic equation.

Example Problem

Sanket bought a rectangular plot. The area is 200 square metres. The length is 10 metres more than the breadth. We need to find the length and breadth.

Let the breadth be x metre.

Then the length is (x + 10) metre.

Area of rectangle = length times breadth

So: 200 = (x + 10) times x

200 = x² + 10x

x² + 10x - 200 = 0

This is a quadratic equation. We need to solve it to find the dimensions of the plot.

Teacher's Note

Real-life problems often give us quadratic equations. When farmers in villages measure their land, they use these equations. Even builders use quadratic equations when planning rectangular plots.

Exam Trick

Always write the equation in standard form: \(ax^2 + bx + c = 0\). This makes it easy to identify a, b, and c. Remember: move everything to one side to get zero on the other side.

Points to Remember

A linear equation has power 1, like 5x + 9.

A quadratic equation has power 2, like \(x^2 + 3x - 5\).

The standard form is \(ax^2 + bx + c = 0\).

Practical problems like finding dimensions use quadratic equations.

The Standard Form Of A Quadratic Equation

The general form of a quadratic equation is \(ax^2 + bx + c = 0\).

In this form, a, b, and c are real numbers. The value of a is never zero (a ≠ 0).

Here a is the coefficient of \(x^2\). Here b is the coefficient of x. Here c is the constant term.

Activity

Complete this table by writing each quadratic equation in standard form.

Teacher's Note

Always move all terms to one side to make the right side zero. For example, if you have \(y^2 = 2y - 7\), rewrite it as \(y^2 - 2y + 7 = 0\). This is how you make the standard form.

Exam Trick

When an equation is given in any form, immediately write it as \(ax^2 + bx + c = 0\). Then pick out a, b, and c. This is the first step in every problem.

Points to Remember

Standard form has all terms on one side and zero on the other.

In the form \(ax^2 + bx + c = 0\), a is never zero.

The coefficient a is the number in front of \(x^2\).

The coefficient b is the number in front of x, and c is the constant.

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